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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St001673
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Mp00294: Standard tableaux —peak composition⟶ Integer compositions
St001673: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001673: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 0
[[1,2]]
=> [2] => 0
[[1],[2]]
=> [2] => 0
[[1,2,3]]
=> [3] => 0
[[1,3],[2]]
=> [3] => 0
[[1,2],[3]]
=> [2,1] => 1
[[1],[2],[3]]
=> [3] => 0
[[1,2,3,4]]
=> [4] => 0
[[1,3,4],[2]]
=> [4] => 0
[[1,2,4],[3]]
=> [2,2] => 0
[[1,2,3],[4]]
=> [3,1] => 1
[[1,3],[2,4]]
=> [3,1] => 1
[[1,2],[3,4]]
=> [2,2] => 0
[[1,4],[2],[3]]
=> [4] => 0
[[1,3],[2],[4]]
=> [3,1] => 1
[[1,2],[3],[4]]
=> [2,2] => 0
[[1],[2],[3],[4]]
=> [4] => 0
[[1,2,3,4,5]]
=> [5] => 0
[[1,3,4,5],[2]]
=> [5] => 0
[[1,2,4,5],[3]]
=> [2,3] => 1
[[1,2,3,5],[4]]
=> [3,2] => 1
[[1,2,3,4],[5]]
=> [4,1] => 1
[[1,3,5],[2,4]]
=> [3,2] => 1
[[1,2,5],[3,4]]
=> [2,3] => 1
[[1,3,4],[2,5]]
=> [4,1] => 1
[[1,2,4],[3,5]]
=> [2,2,1] => 1
[[1,2,3],[4,5]]
=> [3,2] => 1
[[1,4,5],[2],[3]]
=> [5] => 0
[[1,3,5],[2],[4]]
=> [3,2] => 1
[[1,2,5],[3],[4]]
=> [2,3] => 1
[[1,3,4],[2],[5]]
=> [4,1] => 1
[[1,2,4],[3],[5]]
=> [2,2,1] => 1
[[1,2,3],[4],[5]]
=> [3,2] => 1
[[1,4],[2,5],[3]]
=> [4,1] => 1
[[1,3],[2,5],[4]]
=> [3,2] => 1
[[1,2],[3,5],[4]]
=> [2,3] => 1
[[1,3],[2,4],[5]]
=> [3,2] => 1
[[1,2],[3,4],[5]]
=> [2,2,1] => 1
[[1,5],[2],[3],[4]]
=> [5] => 0
[[1,4],[2],[3],[5]]
=> [4,1] => 1
[[1,3],[2],[4],[5]]
=> [3,2] => 1
[[1,2],[3],[4],[5]]
=> [2,3] => 1
[[1],[2],[3],[4],[5]]
=> [5] => 0
[[1,2,3,4,5,6]]
=> [6] => 0
[[1,3,4,5,6],[2]]
=> [6] => 0
[[1,2,4,5,6],[3]]
=> [2,4] => 1
[[1,2,3,5,6],[4]]
=> [3,3] => 0
[[1,2,3,4,6],[5]]
=> [4,2] => 1
[[1,2,3,4,5],[6]]
=> [5,1] => 1
[[1,3,5,6],[2,4]]
=> [3,3] => 0
Description
The degree of asymmetry of an integer composition.
This is the number of pairs of symmetrically positioned distinct entries.
Matching statistic: St000620
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[1,2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0}
[[1],[2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[[1,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[[1,3],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1}
[[1,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1}
[[1,2,3,4]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[[1,3,4],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[[1,2,4],[3]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[[1,2,3],[4]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4,5]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5],[2]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5],[2,4]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5],[3,4]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4],[2,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3],[4,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2,3,4,5,6]]
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5,6],[2,4]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5,6],[3,4]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,6],[2,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,6],[3,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,6],[4,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5],[2,6]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5],[3,6]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd.
The case of an even minimum is [[St000621]].
Matching statistic: St001629
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 67%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [1] => [1] => ? = 0
[[1,2]]
=> [2] => [1] => [1] => ? ∊ {0,0}
[[1],[2]]
=> [2] => [1] => [1] => ? ∊ {0,0}
[[1,2,3]]
=> [3] => [1] => [1] => ? ∊ {0,0,0,1}
[[1,3],[2]]
=> [2,1] => [1,1] => [2] => ? ∊ {0,0,0,1}
[[1,2],[3]]
=> [3] => [1] => [1] => ? ∊ {0,0,0,1}
[[1],[2],[3]]
=> [3] => [1] => [1] => ? ∊ {0,0,0,1}
[[1,2,3,4]]
=> [4] => [1] => [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,3,4],[2]]
=> [2,2] => [2] => [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,2,4],[3]]
=> [3,1] => [1,1] => [2] => ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,2,3],[4]]
=> [4] => [1] => [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,3],[2,4]]
=> [2,2] => [2] => [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,2],[3,4]]
=> [3,1] => [1,1] => [2] => ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,4],[2],[3]]
=> [3,1] => [1,1] => [2] => ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,3],[2],[4]]
=> [2,2] => [2] => [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,2],[3],[4]]
=> [4] => [1] => [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1],[2],[3],[4]]
=> [4] => [1] => [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,2,3,4,5]]
=> [5] => [1] => [1] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5],[2]]
=> [2,3] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5],[3]]
=> [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5],[4]]
=> [4,1] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4],[5]]
=> [5] => [1] => [1] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5],[2,4]]
=> [2,2,1] => [2,1] => [1,1] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5],[3,4]]
=> [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4],[2,5]]
=> [2,3] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4],[3,5]]
=> [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3],[4,5]]
=> [4,1] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4,5],[2],[3]]
=> [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5],[2],[4]]
=> [2,2,1] => [2,1] => [1,1] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5],[3],[4]]
=> [4,1] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4],[2],[5]]
=> [2,3] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4],[3],[5]]
=> [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3],[4],[5]]
=> [5] => [1] => [1] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4],[2,5],[3]]
=> [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3],[2,5],[4]]
=> [2,2,1] => [2,1] => [1,1] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2],[3,5],[4]]
=> [4,1] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3],[2,4],[5]]
=> [2,3] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2],[3,4],[5]]
=> [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,5],[2],[3],[4]]
=> [4,1] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4],[2],[3],[5]]
=> [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3],[2],[4],[5]]
=> [2,3] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2],[3],[4],[5]]
=> [5] => [1] => [1] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1],[2],[3],[4],[5]]
=> [5] => [1] => [1] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4,5,6]]
=> [6] => [1] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5,6],[2]]
=> [2,4] => [1,1] => [2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5,6],[3]]
=> [3,3] => [2] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5,6],[4]]
=> [4,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4,6],[5]]
=> [5,1] => [1,1] => [2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4,5],[6]]
=> [6] => [1] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5,6],[2,4]]
=> [2,2,2] => [3] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,1,1] => [3] => 1
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => [3] => 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,1,1] => [3] => 1
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => [3] => 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,1,1] => [3] => 1
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,1] => [3] => 1
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,1] => [3] => 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,1,1] => [3] => 1
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1] => [3] => 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,1] => [3] => 1
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1] => [3] => 1
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,1,1] => [3] => 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,1] => [3] => 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,1] => [3] => 1
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1] => [3] => 1
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1] => [3] => 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,1] => [3] => 1
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [1,1,1] => [3] => 1
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [1,1,1] => [3] => 1
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [1,1,1] => [3] => 1
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [1,1,1] => [3] => 1
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [1,1,1] => [3] => 1
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [1,1,1] => [3] => 1
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [1,1,1] => [3] => 1
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [1,1,1] => [3] => 1
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [1,1,1] => [3] => 1
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4,7],[2],[5],[6]]
=> [2,4,1] => [1,1,1] => [3] => 1
[[1,3,4,6],[2],[5],[7]]
=> [2,3,2] => [1,1,1] => [3] => 1
[[1,3,6],[2,4,7],[5]]
=> [2,3,2] => [1,1,1] => [3] => 1
[[1,2,5],[3,6,7],[4]]
=> [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4],[2,6,7],[5]]
=> [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4],[2,5,7],[6]]
=> [2,4,1] => [1,1,1] => [3] => 1
[[1,2,3],[4,5,7],[6]]
=> [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4],[2,5,6],[7]]
=> [2,3,2] => [1,1,1] => [3] => 1
[[1,2,7],[3,5],[4,6]]
=> [4,2,1] => [1,1,1] => [3] => 1
[[1,3,7],[2,4],[5,6]]
=> [2,3,2] => [1,1,1] => [3] => 1
[[1,3,6],[2,4],[5,7]]
=> [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4],[2,6],[5,7]]
=> [2,3,2] => [1,1,1] => [3] => 1
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St000526
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000526: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 67%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000526: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [[1],[]]
=> ([],1)
=> ? = 0 + 1
[[1,2]]
=> [2] => [[2],[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[1],[2]]
=> [2] => [[2],[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[1,2,3]]
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1,3],[2]]
=> [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,2],[3]]
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1],[2],[3]]
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[1,2,3,4]]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,3,4],[2]]
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 1 = 0 + 1
[[1,2,4],[3]]
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3],[4]]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,3],[2,4]]
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 1 = 0 + 1
[[1,2],[3,4]]
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[2],[4]]
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 1 = 0 + 1
[[1,2],[3],[4]]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1],[2],[3],[4]]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[1,2,3,4,5]]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1,3,4,5],[2]]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
[[1,2,4,5],[3]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3,5],[4]]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3,4],[5]]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1,3,5],[2,4]]
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[[1,2,5],[3,4]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,3,4],[2,5]]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
[[1,2,4],[3,5]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3],[4,5]]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[1,4,5],[2],[3]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,3,5],[2],[4]]
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[[1,2,5],[3],[4]]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[1,3,4],[2],[5]]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
[[1,2,4],[3],[5]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3],[4],[5]]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1,4],[2,5],[3]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,3],[2,5],[4]]
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[[1,2],[3,5],[4]]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[1,3],[2,4],[5]]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
[[1,2],[3,4],[5]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[1,4],[2],[3],[5]]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[1,3],[2],[4],[5]]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1],[2],[3],[4],[5]]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[1,2,3,4,5,6]]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> [2,4] => [[5,2],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> 2 = 1 + 1
[[1,2,4,5,6],[3]]
=> [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> 1 = 0 + 1
[[1,2,3,5,6],[4]]
=> [4,2] => [[5,4],[3]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> 2 = 1 + 1
[[1,2,3,4,6],[5]]
=> [5,1] => [[5,5],[4]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 1 = 0 + 1
[[1,2,5,6],[3,4]]
=> [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> 1 = 0 + 1
[[1,2,3,4,5,6,7]]
=> [7] => [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,5,6,7],[2]]
=> [2,5] => [[6,2],[1]]
=> ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,5,6,7],[3]]
=> [3,4] => [[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,5,6,7],[4]]
=> [4,3] => [[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4,6,7],[5]]
=> [5,2] => [[6,5],[4]]
=> ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4,5,7],[6]]
=> [6,1] => [[6,6],[5]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4,5,6],[7]]
=> [7] => [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,5,6,7],[2,4]]
=> [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,5,6,7],[3,4]]
=> [3,4] => [[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,6,7],[3,5]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,6,7],[4,5]]
=> [4,3] => [[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4,7],[5,6]]
=> [5,2] => [[6,5],[4]]
=> ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,5,6],[2,7]]
=> [2,5] => [[6,2],[1]]
=> ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,5,6],[3,7]]
=> [3,4] => [[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,5,6],[4,7]]
=> [4,3] => [[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4,6],[5,7]]
=> [5,2] => [[6,5],[4]]
=> ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4,5],[6,7]]
=> [6,1] => [[6,6],[5]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,4,5,6,7],[2],[3]]
=> [3,4] => [[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,5,6,7],[2],[4]]
=> [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,5,6,7],[3],[4]]
=> [4,3] => [[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,6,7],[4],[5]]
=> [5,2] => [[6,5],[4]]
=> ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4,7],[5],[6]]
=> [6,1] => [[6,6],[5]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,5,6],[2],[7]]
=> [2,5] => [[6,2],[1]]
=> ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,5,6],[3],[7]]
=> [3,4] => [[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,5,6],[4],[7]]
=> [4,3] => [[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4,6],[5],[7]]
=> [5,2] => [[6,5],[4]]
=> ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4,5],[6],[7]]
=> [7] => [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,7],[3,5,6]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,7],[4,5,6]]
=> [4,3] => [[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,5,6],[2,4,7]]
=> [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,5,6],[3,4,7]]
=> [3,4] => [[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,6],[3,5,7]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,6],[4,5,7]]
=> [4,3] => [[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4],[5,6,7]]
=> [5,2] => [[6,5],[4]]
=> ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
Description
The number of posets with combinatorially isomorphic order polytopes.
Matching statistic: St000260
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => ([],1)
=> 0
[[1,2]]
=> [2] => ([],2)
=> ? ∊ {0,0}
[[1],[2]]
=> [2] => ([],2)
=> ? ∊ {0,0}
[[1,2,3]]
=> [3] => ([],3)
=> ? ∊ {0,0,0}
[[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[[1,2],[3]]
=> [3] => ([],3)
=> ? ∊ {0,0,0}
[[1],[2],[3]]
=> [3] => ([],3)
=> ? ∊ {0,0,0}
[[1,2,3,4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0}
[[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0}
[[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,2,3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0}
[[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0}
[[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0}
[[1,2],[3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0}
[[1],[2],[3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0}
[[1,2,3,4,5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5],[2]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,5],[3,4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4],[2,5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,4,5],[2],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,3,4],[2],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3],[4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4],[2,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,3],[2,4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2],[3,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,4],[2],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3],[2],[4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1],[2],[3],[4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5,6],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5,6],[2,4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5,6],[3,4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,6],[2,5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,3,6],[4,5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5],[2,6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,4,5,6],[2],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5,6],[2],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5,6],[3],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,6],[2],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,4,5],[2],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4],[5],[6]]
=> [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5],[3,4,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4],[2,5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,4],[3,5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,4,6],[2,5],[3]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,6],[3,4],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,4],[3,6],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,3],[4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2],[3,4],[5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2],[3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,6],[2],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,3,4,5],[6,7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 67%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [1] => ([],1)
=> ? = 0 - 1
[[1,2]]
=> [2] => [1,1] => ([(0,1)],2)
=> -1 = 0 - 1
[[1],[2]]
=> [2] => [1,1] => ([(0,1)],2)
=> -1 = 0 - 1
[[1,2,3]]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[[1,3],[2]]
=> [2,1] => [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
[[1,2],[3]]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[[1],[2],[3]]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[[1,2,3,4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[[1,3,4],[2]]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0} - 1
[[1,2,4],[3]]
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,2,3],[4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[[1,3],[2,4]]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0} - 1
[[1,2],[3,4]]
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,4],[2],[3]]
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[2],[4]]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0} - 1
[[1,2],[3],[4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[[1],[2],[3],[4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 0 - 1
[[1,3,4,5],[2]]
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,4,5],[3]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,3,5],[4]]
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,3,4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 0 - 1
[[1,3,5],[2,4]]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,5],[3,4]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,3,4],[2,5]]
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,4],[3,5]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,3],[4,5]]
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4,5],[2],[3]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,3,5],[2],[4]]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,5],[3],[4]]
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,3,4],[2],[5]]
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,4],[3],[5]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,3],[4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 0 - 1
[[1,4],[2,5],[3]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,3],[2,5],[4]]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2],[3,5],[4]]
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,3],[2,4],[5]]
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2],[3,4],[5]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,5],[2],[3],[4]]
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4],[2],[3],[5]]
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,3],[2],[4],[5]]
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 0 - 1
[[1],[2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 0 - 1
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 0 - 1
[[1,3,4,5,6],[2]]
=> [2,4] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,4,5,6],[3]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,3,5,6],[4]]
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,3,4,6],[5]]
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,4,5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 0 - 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,5,6],[3,4]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,3,6],[4,5]]
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,3,4,5],[2,6]]
=> [2,4] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,4,5],[3,6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,3,5],[4,6]]
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,3,4],[5,6]]
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,4,5,6],[2],[3]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,5,6],[3],[4]]
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,3,6],[4],[5]]
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,4,5],[2],[6]]
=> [2,4] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,4,5],[3],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,3,5],[4],[6]]
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,3,4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 0 - 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,5],[3,4,6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,3],[4,5,6]]
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,6],[3,5],[4]]
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,4,5],[2,6],[3]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} - 1
[[1,2,3],[4,6],[5]]
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,6],[3],[4],[5]]
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 0 - 1
[[1,2],[3,6],[4],[5]]
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,6],[2],[3],[4],[5]]
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2],[3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 0 - 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 0 - 1
[[1,2,3,4,5,6,7]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> -1 = 0 - 1
[[1,2,3,4,5,7],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,5,6],[7]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> -1 = 0 - 1
[[1,2,3,4,5],[6,7]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,7],[5],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,5],[6],[7]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> -1 = 0 - 1
[[1,2,3,4],[5,7],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,7],[4],[5],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4],[5],[6],[7]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> -1 = 0 - 1
[[1,2,3],[4,7],[5],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,7],[3],[4],[5],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3],[4],[5],[6],[7]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> -1 = 0 - 1
[[1,2],[3,7],[4],[5],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,7],[2],[3],[4],[5],[6]]
=> [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001846
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [1] => ([(0,1)],2)
=> 0
[[1,2]]
=> [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 0
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 0
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 0
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {0,1}
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 0
[[1,2,3],[4]]
=> [4,1,2,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 0
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 0
[[1,2],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {0,1}
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [5,3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,4,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,5,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 0
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,5,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,1,2,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,10),(5,11),(5,12),(6,9),(6,13),(8,10),(9,7),(10,9),(11,8),(12,8),(13,7)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [5,1,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [4,2,1,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [4,3,2,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,11),(4,14),(5,12),(5,15),(6,13),(6,15),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,7),(15,9),(15,10)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [5,2,1,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,9),(5,11),(6,7),(6,10),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [5,3,2,1,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [5,4,2,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,14),(3,14),(4,11),(5,7),(6,12),(6,13),(8,10),(9,10),(10,7),(11,9),(12,8),(13,8),(13,9),(14,11),(14,13)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [6,2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [6,3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,7),(5,10),(5,13),(6,11),(6,12),(8,10),(9,7),(10,9),(11,8),(12,8),(13,9)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [6,4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,10),(4,9),(5,8),(6,11),(7,9),(7,10),(9,12),(10,12),(11,8),(12,11)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [6,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4,1,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [3,4,2,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [2,5,1,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [3,5,2,1,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [4,5,2,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,11),(9,12),(11,12),(12,10)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [2,6,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [3,6,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [4,6,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [5,6,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,9),(4,12),(5,12),(6,10),(6,11),(8,7),(9,7),(10,13),(11,13),(12,8),(13,8),(13,9)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [6,2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6,4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [2,5,3,1,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 0
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [2,6,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 0
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [2,6,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 0
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,4,6,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 0
[[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => [3,6,4,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 0
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [5,3,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 0
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [3,6,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 0
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 0
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [3,5,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 0
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [4,6,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 0
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [3,5,6,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 0
[[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => [3,7,5,2,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [3,5,7,2,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [5,3,1,7,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [6,4,1,7,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0
[[1,4,6],[2,5],[3],[7]]
=> [7,3,2,5,1,4,6] => [3,5,1,7,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [4,7,1,6,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0
[[1,3,5],[2,4],[6],[7]]
=> [7,6,2,4,1,3,5] => [4,6,1,7,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [3,5,7,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0
[[1,5],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5] => [4,6,3,1,7,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0
[[1,4],[2,6],[3,7],[5]]
=> [5,3,7,2,6,1,4] => [5,7,3,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0
[[1,4],[2,6],[3],[5],[7]]
=> [7,5,3,2,6,1,4] => [3,5,7,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0
Description
The number of elements which do not have a complement in the lattice.
A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.
Matching statistic: St001820
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001820: Lattices ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001820: Lattices ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [1] => ([(0,1)],2)
=> 1 = 0 + 1
[[1,2]]
=> [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {0,0} + 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 1 + 1
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[[1,2,3],[4]]
=> [4,1,2,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 1 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {0,0} + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 1 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 0 + 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [5,3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 0 + 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,4,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,5,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,5,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,1,2,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,10),(5,11),(5,12),(6,9),(6,13),(8,10),(9,7),(10,9),(11,8),(12,8),(13,7)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [5,1,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [4,2,1,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [4,3,2,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,11),(4,14),(5,12),(5,15),(6,13),(6,15),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,7),(15,9),(15,10)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [5,2,1,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,9),(5,11),(6,7),(6,10),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [5,3,2,1,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [5,4,2,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,14),(3,14),(4,11),(5,7),(6,12),(6,13),(8,10),(9,10),(10,7),(11,9),(12,8),(13,8),(13,9),(14,11),(14,13)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [6,2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [6,3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,7),(5,10),(5,13),(6,11),(6,12),(8,10),(9,7),(10,9),(11,8),(12,8),(13,9)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [6,4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,10),(4,9),(5,8),(6,11),(7,9),(7,10),(9,12),(10,12),(11,8),(12,11)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [6,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4,1,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [3,4,2,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [2,5,1,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [3,5,2,1,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [4,5,2,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,11),(9,12),(11,12),(12,10)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [2,6,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [3,6,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [4,6,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [5,6,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,9),(4,12),(5,12),(6,10),(6,11),(8,7),(9,7),(10,13),(11,13),(12,8),(13,8),(13,9)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [6,2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6,4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [2,5,3,1,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 0 + 1
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [2,6,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [2,6,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,4,6,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 0 + 1
[[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => [3,6,4,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [5,3,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 0 + 1
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [3,6,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 0 + 1
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [3,5,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 0 + 1
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [4,6,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [3,5,6,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => [3,7,5,2,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [3,5,7,2,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [5,3,1,7,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [6,4,1,7,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[1,4,6],[2,5],[3],[7]]
=> [7,3,2,5,1,4,6] => [3,5,1,7,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [4,7,1,6,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[1,3,5],[2,4],[6],[7]]
=> [7,6,2,4,1,3,5] => [4,6,1,7,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [3,5,7,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[1,5],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5] => [4,6,3,1,7,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[1,4],[2,6],[3,7],[5]]
=> [5,3,7,2,6,1,4] => [5,7,3,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[1,4],[2,6],[3],[5],[7]]
=> [7,5,3,2,6,1,4] => [3,5,7,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
Description
The size of the image of the pop stack sorting operator.
The pop stack sorting operator is defined by $Pop_L^\downarrow(x) = x\wedge\bigwedge\{y\in L\mid y\lessdot x\}$. This statistic returns the size of $Pop_L^\downarrow(L)\}$.
Matching statistic: St001570
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[[1]]
=> [1] => ([],1)
=> ([],1)
=> ? = 0
[[1,2]]
=> [2] => ([],2)
=> ([],1)
=> ? ∊ {0,0}
[[1],[2]]
=> [2] => ([],2)
=> ([],1)
=> ? ∊ {0,0}
[[1,2,3]]
=> [3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,1}
[[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1}
[[1,2],[3]]
=> [3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,1}
[[1],[2],[3]]
=> [3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,1}
[[1,2,3,4]]
=> [4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,2,3],[4]]
=> [4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,2],[3],[4]]
=> [4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1],[2],[3],[4]]
=> [4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1}
[[1,2,3,4,5]]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5],[2]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4],[5]]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,5],[3,4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4],[2,5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4,5],[2],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4],[2],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3],[4],[5]]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4],[2,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3],[2,4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2],[3,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4],[2],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3],[2],[4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1],[2],[3],[4],[5]]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5,6],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5,6],[2,4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,5,6],[3,4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,6],[2,5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,3,6],[4,5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5],[2,6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5,6],[2],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,4,6],[2],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,4],[2,5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,4],[3,5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,4,6],[2,5],[3]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,6],[2,5],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,6],[2,4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,6],[3,4],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,5],[2,6],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,4],[2,6],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,4],[3,6],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,5],[2,4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3],[2,5],[4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3],[2,4],[5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2],[3,4],[5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St001868
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [1] => [1] => 0
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [2,1] => [2,1] => 0
[[1,2,3]]
=> [1,2,3] => [3,2,1] => [3,2,1] => 0
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => [3,1,2] => 0
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [4,2,3,1] => [4,2,3,1] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [4,1,2,3] => [4,1,2,3] => 0
[[1,2,4],[3]]
=> [3,1,2,4] => [2,4,3,1] => [2,4,3,1] => 0
[[1,2,3],[4]]
=> [4,1,2,3] => [3,4,2,1] => [3,4,2,1] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [4,3,2,1] => [4,3,2,1] => 0
[[1,2],[3,4]]
=> [3,4,1,2] => [2,3,1,4] => [2,3,1,4] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,4,2] => [1,3,4,2] => 0
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,2,1,4] => [3,2,1,4] => 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,2,3,4,1] => [5,2,3,4,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [5,1,2,3,4] => [5,1,2,3,4] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,5,3,4,1] => [2,5,3,4,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [3,5,2,4,1] => [3,5,2,4,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [4,5,2,3,1] => [4,5,2,3,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [2,3,5,4,1] => [2,3,5,4,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [5,4,2,3,1] => [5,4,2,3,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [2,4,5,3,1] => [2,4,5,3,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,4,5,2,1] => [3,4,5,2,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [4,3,5,2,1] => [4,3,5,2,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [2,5,4,3,1] => [2,5,4,3,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [3,5,4,2,1] => [3,5,4,2,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [3,2,4,1,5] => [3,2,4,1,5] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [4,5,3,2,1] => [4,5,3,2,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [4,2,3,1,5] => [4,2,3,1,5] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,4,2,5,3] => [1,4,2,5,3] => 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,4,3,5,2] => [1,4,3,5,2] => 0
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,2,3,4,5,1] => [6,2,3,4,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [2,6,3,4,5,1] => [2,6,3,4,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [3,6,2,4,5,1] => [3,6,2,4,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [4,6,2,3,5,1] => [4,6,2,3,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [5,6,2,3,4,1] => [5,6,2,3,4,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [6,3,2,4,5,1] => [6,3,2,4,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [2,3,6,4,5,1] => [2,3,6,4,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [6,4,2,3,5,1] => [6,4,2,3,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [2,4,6,3,5,1] => [2,4,6,3,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [3,4,6,2,5,1] => [3,4,6,2,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [6,5,2,3,4,1] => [6,5,2,3,4,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [2,5,6,3,4,1] => [2,5,6,3,4,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [3,5,6,2,4,1] => [3,5,6,2,4,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [4,5,6,2,3,1] => [4,5,6,2,3,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [1,3,6,2,4,5] => [1,3,6,2,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [3,2,6,4,5,1] => [3,2,6,4,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [1,4,6,2,3,5] => [1,4,6,2,3,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [4,2,6,3,5,1] => [4,2,6,3,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [4,3,6,2,5,1] => [4,3,6,2,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [1,5,6,2,3,4] => [1,5,6,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [5,2,6,3,4,1] => [5,2,6,3,4,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [5,3,6,2,4,1] => [5,3,6,2,4,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [5,4,6,2,3,1] => [5,4,6,2,3,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [6,3,5,2,4,1] => [6,3,5,2,4,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [2,3,5,6,4,1] => [2,3,5,6,4,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [6,4,5,2,3,1] => [6,4,5,2,3,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [2,4,5,6,3,1] => [2,4,5,6,3,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [3,4,5,6,2,1] => [3,4,5,6,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [2,6,4,3,5,1] => [2,6,4,3,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The number of alignments of type NE of a signed permutation.
An alignment of type NE of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1 \leq i, j\leq n$ such that $\pi(i) < i < j \leq \pi(j)$.
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